Population dynamics in heterogeneous conditions

Using a Monte Carlo approach we study a simple lattice model of populations living in a habitat where the external conditions are changing in space and in time. We show that above a certain value of the climatic gradient, the population gathers at a restricted part of the lattice. The average trait of the individuals follows the optimum. We also have shown that there exists a range of gradient values within which a population has over 10% chance of survival, while outside it most likely to become extinct. We have found that this phenomenon depends on the selection pressure and we have constructed a phase diagram in the selection-gradient plane. In the case of the time-dependent optimum, the populations go extinct after a long time, depending on the speed of the changes.